1 Introduction

The investigation of non-Newtonian fluids is extensive subject of extreme interest for the scientists and experts. The non-Newtonian fluid is defined as the nonlinear relations between shear rate and shear stress. The non-Newtonian fluids are most regularly used in blood, printing colours and inks, greasing oils and Greases, toothpaste, shampoos, nail polish, cosmetics, and creams and lotions, shaving suds, snacks, combination of cake and coatings cake, white egg, dressings salad, jams, bread amalgams various molten polymers are to mention just a few with daily life concern. Various novelists examine different kinds of non-Newtonian fluid models are deliberating their flow in the direction of different surfaces in an occurrence of an appropriate physical effects. The historical and latest developments in this direction can be assessed in Refs. [1,2,3,4,5]. The materials specifically melting proteins and polymers, suds, solid creams, waxy oils, polymers solution, crudes oil, pharmaceuticals reproduces the thinning of shear characteristics. For such kinds of non-Newtonian fluid constituents, the influence of viscosity is initiate to be an occupation of shear rate and can be designated by an appropriate fluid model similar to Carreau fluid models [6, 7]. Three-dimensional flow of Casson–Carreau fluids stretching surface model deliberated by Raju et al. and Gireesha et al. [8, 9]. The stratification model an appropriate to the period of non-Newtonian fluid model is deliberated by investigators [10,11,12,13,14,15], here Carreau fluid model delivers enhanced estimates for many shear thinning fluids for the constitutive association of Carreau fluid model grips at both lower and higher shear rates.

The property of Magneto hydrodynamic (MHD) is the electrically magnifying the fluids, and then a conductor transporting electrical energy exchanges into a magnetic field. Magneto hydrodynamic (MHD) has numerous significances in magnetic sensors, cosmology, astrophysics, and geophysics, drugs, engineering and solid. The impact of heat transfer in terms of thermal radiation and MHD assume a noteworthy part in industry especially in assembling and industrial equipment’s that incorporate rockets, space vehicles, gas turbines, satellites, nuclear plants and many others. So many researchers studied on MHD Carreau fluid flow [16,17,18,19,20]. The stratification impact on magneto hydrodynamics fluid flow is founded in the poetry as yet particularly when fluid flow is encouraged by stretching cylindrical surface are discoursed Khalil et al. [21], Mixed Convection Stagnation-Point Flow on MHD discussed by Madhu et al. [22], heat transfer over stretching sheet variable viscous dissipation and heat source/sink on MHD has deliberated by Dessie [23]. Non-Newtonian power-law fluids flowing over a wedge with heat source/sink in presence of viscous dissipation deliberated by Kishan et al. [24], Here, the uniqueness of current effort is to study the velocity, temperature nanofluid stratification enforced in the fluid flow system of magneto hydrodynamic (MHD) Carreau fluid flow models. Recently, the various investigators [25,26,27,28,29,30,31,32,33,34,35,36] are considered magneto hydrodynamic with the various flow configurations (disk, stretched surface, cylinder, symmetric and asymmetric flow) and various physical parameters (suction, Multiwall carbon nanotubes, nanoparticles, non-Fourier heat flux). With their studies highlighted that the non-Newtonian fluid has tendency to control the flow performance.

The ignoring on overwhelming standing of generation/absorption reactions on MHD Carreau fluid flow with thermally stratified and the boundary layer flow of MHD Carreau fluid over a permeable cylinder. The major objective of this work is to scrutinize the impacts of generation/absorption and thermally reactions on MHD Carreau fluid flow over a linearly permeable cylinder under the inspiration of functional magnetic field. The fluid flow of governing equations is numerically attempted with shooting technique. The consequence of governing physical parameters are interested quantities (velocity, temperature, nanofluid, skin friction, Nusselt and Sherwood numbers) are specified through graphical and tabular illustrations.

2 Mathematical formulation

The steady 2-dimensional boundary layer flow of incompressible thermally stratified Carreau fluid flow over a permeable stretching cylinder surface has been deliberated. The coordinate system is elected in such a method that x-axis is restrained along with the stretching cylinder though r-axis. The magnetic field \((B_{0} )\) is identical and functional in r-direction and the encouraged magnetic field is under small Reynolds magnetic number. The stretching cylinder velocity is supposed to be \(u_{w} = U_{0} xL^{ - 1}\) where \(U_{0} > 0\), and \(L\) is the specific length. The surface temperature of the cylinder is \(T_{w} (x) = T_{0} + bxL^{ - 1} ,\) where \(b\) is arbitrary constant and \(T_{0}\) is the free stream temperature. Here, Carreau fluid flow has been expected alongside x-direction and r-direction is measured perpendicular to fluid flow.

Also taking the two dimensional cylindrical steady flow, velocity, temperature, nanoparticle volume friction are theoretical to be the form of:

$$V = \left[ {u = \left( {x, \, r} \right); \, v = \left( {x,r} \right)} \right],T = T\left( {x,r} \right),C = C\left( {x,r} \right)$$
(1)

Hear, \(u\) and \(v\) denotes the velocity variable in \(x -\) and \(r -\) direction. The convective model transport designed for the Carreau fluid flow model is governing by the equations of mass, momentum, energy and concentration are followed as [16]:

$$(ru)_{x} + (rv)_{r} = 0$$
(2)
$$u(u_{x} ) + v(u_{r} ) \,=\, \upsilon^{\prime} \left[ {u_{rr} + r^{ - 1} u_{r} + \frac{{3\varGamma^{2} (n - 1)}}{2}(u_{r} )^{2} (u_{rr} ) + \frac{{3\varGamma^{2} r^{ - 1} (n - 1)}}{2}(u_{r} )^{3} } \right] - \sigma B_{0}^{2} \rho^{ - 1} u$$
(3)
$$u(T_{x} ) + v(T_{r} ) = \alpha^{\prime} \left[ {T_{rr} } \right. + \left. {r^{ - 1} T_{r} } \right] + \tau^{*} \left[ {D_{B} } \right.C_{r} T_{r} + \left. {D_{T} (T_{r} )^{2} } \right] + Q_{0} (\rho cp)^{ - 1} (T - T_{\infty } )$$
(4)
$$u(C_{x} ) + v(C_{r} ) = D_{B} \left[ {C_{rr} } \right. + \left. {r^{ - 1} C_{r} } \right] + D_{T} T_{\infty }^{ - 1} T_{rr}$$
(5)

In above equations \(u\) and \(v\) are the velocity components. These components are parallel to \(x\) and \(r\) directions correspondingly. Here kinematic viscosity is \(\upsilon^{\prime}\), the electrical conductivity of the fluid is \(\sigma\), the coefficient of thermal expansion is \(\beta_{T}\), the gravity is \(g\), the fluid density is \(\rho_{f}\), the fluid temperature is \(T\), the temperature of the ambient fluid is \(T_{\infty }\), the coefficient of expansion with concentration is \(\beta_{C}\), the fluid concentration is \(C\) and the nanoparticle concentration \(C_{\infty }\) is far away from the surface, the thermal diffusivity is \(\alpha^{\prime}\), the effective heat capacity of the nanoparticle is \((\rho c)_{p}\) and the effective heat capacity of fluid is \((\rho c)_{f}\), the Brownian diffusion coefficient is \(D_{B}\), the thermophoresis diffusion coefficient is \(D_{T}\), the specific heat capacity is \(c_{p}\), the volumetric heat generation/absorption is \(Q_{0}\).

Also, the boundary condition are given below:

$$\begin{aligned} & u(x,r) = U_{0} L^{ - 1} x,\,v(x,r) = 0,\,T(x,r) = T_{w} (x) = T_{0} + bxL^{ - 1} ,\,C = C_{w} (x)\;{\text{at}}\;r = R \\ & u(x,r) \to 0,\,v(x,r) \to 0,\,T(x,r) \to T_{\infty } (x) = T_{0} + cxL^{ - 1} ,\,C(x,r) = C_{\infty } (x)\;{\text{as}}\;r \to \infty \\ \end{aligned}$$
(6)

The velocity components in terms of stream function \(\psi\) can be written by form

$$u(x,r) = r^{ - 1} (\psi_{r} ),\,v = - r^{ - 1} (\psi_{x} )$$

Where \(\psi_{r} ,\,\,\psi_{x}\) are differentiation with respect to \(r,\,x\) and similarity variables are

$$\begin{aligned} u & = U_{0} xL^{ - 1} F^{\prime} (\varsigma ),\,v = - Rr^{ - 1} \left[ {U_{0} \upsilon^{*} L^{ - 1} } \right]^{0.5} ,\quad \varsigma = (2R)^{ - 1} \left[ {r^{2} - R^{2} (U_{0} (\upsilon^{*} L)^{ - 1} )^{ - 0.5} } \right], \\ \psi & = \left[ {U_{0} \upsilon^{*} x^{2} L^{ - 1} } \right]^{ - 0.5} ,\,(T_{w} - T_{0} )T(\varsigma ) = T - T_{\infty } ,\,(C_{w} - C_{0} )C(\varsigma ) = C - C_{\infty } \\ \end{aligned}$$
(7)

The above governing Eqs. (2)–(5) are transformed as follows,

$$\begin{aligned} & (1 + 2K\varsigma )F_{\varsigma \varsigma \varsigma } (\varsigma )\left[ {1 + \frac{3}{2}(n - 1)\lambda^{2} (1 + 2K\varsigma )(F_{\varsigma \varsigma } (\varsigma ))^{2} } \right] + KF_{\varsigma \varsigma } (\varsigma )\left[ {1 + (1 + 2K\varsigma )\frac{n - 1}{2}\lambda^{2} F_{\varsigma \varsigma }^{2} } \right] \\ & \quad + F(\varsigma )F_{\varsigma \varsigma } (\varsigma ) - F_{\varsigma } (\varsigma )^{2} - MF_{\varsigma } (\varsigma ) = 0 \\ \end{aligned}$$
(8)
$$(1 + 2K\varsigma )T_{\varsigma \varsigma } (\varsigma ) + KT_{\varsigma } + \Pr (1 + 2K\varsigma )\left[ {NbT(\varsigma )C_{\varsigma } (\varsigma ) + NtT_{\varsigma } (\varsigma )^{2} } \right] + F(\varsigma )T_{\varsigma } (\varsigma ) + HT(\varsigma ) = 0$$
(9)
$$(1 + 2K\varsigma )C_{\varsigma \varsigma } (\varsigma ) + KC_{\varsigma } (\varsigma ) + Le\Pr F(\varsigma )C_{\varsigma } (\varsigma ) + (1 + 2K\varsigma )\frac{Nt}{Nb}T_{\varsigma \varsigma } (\varsigma ) = 0$$
(10)

The non-dimensional form of the boundary conditions are,

$$\begin{aligned} & F(\varsigma ) = 0,\,F_{\varsigma } (\varsigma ) = 1,\quad T(\varsigma ) = 1 - k_{1} ,\,C(\varsigma ) = 1,\,\;{\text{as}}\;\varsigma \to 0 \\ & F_{\varsigma } (\varsigma ) \to 0,\,T(\varsigma ) \to 0,\quad C(\varsigma ) \to 0\;{\text{at}}\;\varsigma \to \infty \\ \end{aligned}$$
(11)

In above equation, \(F_{\varsigma }\) denotes differentiation with respect to \(\varsigma\). The non-dimensional parameters defined as:

Where some of non-dimensional parameters associated as, \(n\) is power law index, the curvature parameter is \(K\left( { = \frac{1}{R}\sqrt {\frac{{L\upsilon^{*} }}{{U_{0} }}} } \right)\), the thermal stratification parameter is \(k_{1} \left( { = \frac{c}{b}} \right)\), the Weissenberg number is \(\lambda \left( { = \varGamma \sqrt {\frac{{U_{0}^{2} }}{{L^{2} \upsilon^{*} }}} x} \right)\), the magnetic parameter is \(M\left( { = \frac{{2\sigma B_{0}^{2} }}{{a\rho_{f} (n + 1)}}} \right)\), the Prandtl number is \(\,\Pr ( = \upsilon /\alpha )\), the Brownian motion parameter is \(Nb\left( { = \frac{{(\rho c)_{p} D_{B} (C_{w} - C\infty )}}{{\upsilon (\rho c)_{f} }}} \right)\), the thermophoresis parameter is \(Nt\left( { = \frac{{(\rho c)_{p} D_{T} (T_{w} - T_{\infty } )}}{{\upsilon (\rho c)_{f} T_{\infty } }}} \right)\), and the heat generation/absorption coefficient is \(H\left( { = \frac{{2xQ_{0} }}{{(\rho c)_{f} (n + 1)U_{w} }}} \right)\).

For engineering concerned, physical parameters are the skin fraction coefficient \(C_{{f_{x} }}\), local Nusselt number \(Nu_{x}\) and local Sherwood number \(Sh_{x}\) given as,

$$\begin{aligned} C_{F} & = \frac{{T_{w} }}{{\rho U_{w}^{2} }}\;{\text{where}}\;\tau_{w} = \mu \left[ {u_{r} + \frac{{\varGamma^{2} (n - 1)}}{\sqrt 2 }(u_{r} )^{3} } \right]_{r = R} , \\ Nu_{x} & = \frac{{xq_{w} }}{{k(T_{w} - T_{0} )}}\;{\text{where}}\;q_{w} = - k(T_{r} )_{r = R} , \\ Sh_{x} & = \frac{{xq_{m} }}{{D_{B} (C_{w} - C_{0} )}}\;{\text{where}}\;q_{m} = - D_{m} (C_{r} )_{r = R} . \\ \end{aligned}$$
(12)

Ultimately by introducing Eq. (7) the above Eq. (12) is converting into surface drag force, heat transfer and mass transfer rates are

$$\text{Re}_{x}^{0.5} C_{F} = F_{\varsigma \varsigma } (0) + \frac{{\lambda^{2} (n - 1)}}{2}(F_{\varsigma \varsigma } (0))^{3} ,\,\text{Re}_{x}^{ - 0.5} Nu_{x} = - T_{\varsigma } (0),\,\text{Re}_{x}^{ - 0.5} Sh_{x} = - C_{\varsigma } (0)$$

\(\text{Re}_{x} = \frac{{xU_{0} x^{2} }}{{\upsilon^{*} L}}\) is shows that Reynolds number.

3 Numerical solution

In this section, the governing Eqs. (8)–(10) with boundary condition (11) are reduced to IVP by decreasing the higher order into first order with using dummy substitution is achieved as follows

$$\left( \begin{aligned} u_{1} \hfill \\ u_{2} \hfill \\ u_{3} \hfill \\ u_{4} \hfill \\ u_{5} \hfill \\ u_{6} \hfill \\ u_{7} \hfill \\ \end{aligned} \right) = \left( \begin{aligned}& F(\varsigma ) \hfill \\ &(u_{1} ) = F_{\varsigma } (\varsigma ) \hfill \\& (u_{2} )_{\varsigma } = (u_{1} )_{\varsigma \varsigma } = F_{\varsigma \varsigma } (\varsigma ) \hfill \\ &T(\varsigma ) \hfill \\& (u_{4} )_{\varsigma } = T_{\varsigma } (\varsigma ) \hfill \\& C(\varsigma ) \hfill \\& (u_{6} )_{\varsigma } = C_{\varsigma } (\varsigma ) \hfill \\ \end{aligned} \right)$$

Replace the above given equation are the governing equations and boundary conditions (8)–(11) are solved by numerically via shooting method. Additionally, it is important to select a finite value of \(\varsigma_{\infty }\). The higher order equations converted as follows

$$\left( \begin{aligned} (u_{1} )_{\varsigma } \hfill \\ (u_{2} )_{\varsigma } \hfill \\ (u_{3} )_{\varsigma } \hfill \\ (u_{4} )_{\varsigma } \hfill \\ (u_{5} )_{\varsigma } \hfill \\ (u_{6} )_{\varsigma } \hfill \\ (u_{7} )_{\varsigma } \hfill \\ \end{aligned} \right) = \left( \begin{aligned} &u_{2} \hfill \\ &u_{3} \hfill \\ &\frac{{(u_{2} )^{2} + Mu_{2} - u_{1} u_{3} - Ku_{3} \left[ {1 + (n - 1)/2(1 + 2K\varsigma )\lambda^{2} u_{3}^{2} } \right]}}{{(1 + 2K\varsigma )(1 + 1.5(n - 1)\lambda^{2} (1 + 2K\varsigma )u_{3}^{2} )}} \hfill \\ & u_{5} \hfill \\ &- \frac{{Ku_{5} + \Pr (1 + 2K\varsigma )(Nbu_{4} u_{7} + Ntu_{5}^{2} + u_{1} u_{5} + Hu_{4} )}}{(1 + 2K\varsigma )} \hfill \\& u_{6} \hfill \\ &- \frac{{Ku_{7} + Le\Pr u_{1} u_{7} + (Nt/Nb)(1 + 2K\varsigma )u_{5} }}{(1 + 2K\varsigma )} \hfill \\ \end{aligned} \right)$$

In the above \((u_{r} )_{\varsigma } ,\,r = 1\) to 7 denotes differentiation with respect to \(\varsigma\). The converted boundary conditions in relation of one point conditions are written below

$$u_{1} (0) = 0,\,u_{2} (0) = 1,\,\,u_{3} (0) = h_{1} ,\,\,u_{4} (0) = 1 - k_{1} ,\,\,u_{5} (0) = h_{2} ,\,\,u_{6} (0) = 1,\,\,u_{7} (0) = h_{3.}$$

The outline field condition are converted as\(\,u_{2} (\infty ) = 0,\,\,u_{4} (\infty ) = 0,\,u_{6} (\infty ) = 0\) where \(h_{1} ,\,h_{2}\) and \(h_{3}\) are primary used on the performance of computational algorithm.

4 Results and discussion

For the observation results in the current study a numerical computation is achieved for the impact of heat generation/absorption and thermally stratified on MHD Carreau nanofluid on a permeable cylinder. The non-linear coupled governing Eqs. (8)–(10) and the equivalent boundary condition Eq. (11) are achieved by numerically applying shooting technique. Additionally, characteristic outcomes of skin friction \((C_{F} )\), Nusselt number \((Nu)\) and Sherwood number \((Sh)\) are chronicled with the tables. The various non-dimensional identical parameters \(\lambda ,M, \, n,H^{ + } ,\,H^{ - } ,Nb,Nt,K, \, Pr, \, Le\) on the flow, dimensionless fluid velocity \(F^{\prime} (\varsigma )\), temperature \(T(\varsigma )\) and Nanofluid concentration \(C(\varsigma )\) profiles are displayed through graphs. The variation in the velocity, temperature, nanoparticle volume fraction profiles for different values of Weissenberg number \(\left( {\lambda = \, 1.2, \, 1.4,1.6} \right)\) for both the power law index of values at \(n < 1\) shear thinning and \(n > 1\) shear thickening cases, as it is noticed from the figures that the shear thinning flow case has lesser than the shear thickening flow case. Figures 1, 2 and 3 display that the velocity, temperature, nanoparticle volume fraction profiles have a thickness of thermal boundary layers moderate with enriching values of Weissenberg number in both cases of shear thickening and shear thinning fluids. Here, we found that the higher variations in the velocity, temperature, nanoparticle volume fraction for improved values of Weissenberg number, whereas the temperature, nanoparticle volume fraction profiles are boosting with boosting of Weissenberg number and velocity decreases.

Fig. 1
figure 1

Influence of Weissenberg number \((\lambda )\) on velocity profile \((F^{\prime}(\varsigma ))\)

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Fig. 2
figure 2

Influence of Weissenberg number \((\lambda )\) on temperature profile \((T(\varsigma ))\)

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Fig. 3
figure 3

Influence of Weissenberg number \((\lambda )\) on nano particle volume fraction profile \((C(\varsigma ))\)

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Figure 4 shows the performance of the velocity profile for several values of physical parameters. Imitations of curvature parameter and buoyancy forces are floating body in cylinder flow has two different value of magnetic parameter. In this figure curvature parameter \(\left( {K = \, 1.3, \, 1.6,1.9} \right)\) values are boosting then the velocity profile declining, the impression of buoyancy parameter has moreover controlling than the curvature parameter. Figure 5 exposes that impression of magnetic field parameter \((M)\) on nanoparticle volume friction \(C(\varsigma )\) profiles for two specific changed flows of power law index at \(n < 1\) it shear thinning and \(n > 1\) it shear thickening respected the circulation, at shear thinning flow case is reduces then the shear thickening flow cases. The clear improvement of magnetic parameter \(\left( {M \, = \, 1.0,\,3.0, \, 5.0} \right)\) enhancing nanoparticle volume friction profile due to the impeding force on the flow and rising on magnetic parameter M increase Lorentz force produced by magnetic field, thus degeneration struggle to the nanoparticle fluid flow.

Fig. 4
figure 4

Influence of curvature parameter \((K)\) on velocity profile \((F^{\prime}(\varsigma ))\)

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Fig. 5
figure 5

Influence of magnetic parameter \((M)\) on nano particle volume fraction profile \((C(\varsigma ))\)

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Figure 6 shows the impact of Brownian motion parameter \((Nb)\) on velocity profile for two cases of generation and absorption flows. The influence of Brownian motion is higher in heat generation case compared with heat absorption case. Moreover, the boosting values of Brownian motion \(\left( {Nb = \, 1.1,\,1.3,{ 1}.5} \right)\) enhanced the velocity profile in both the heat generation and absorption flow cases. Figure 7, expresses the concentration profiles in the presence of Brownian motion for both the magnetic and without magnetic field cases. The concentration profiles are reduced with improvement in the Brownian motion. Figures 8, 9 and 10 demonstrate that the various values of thermophoresis parameters \(\left( {Nt = \, 1,\,2,3} \right)\) affecting velocity, temperature and nanoparticle volume fraction profiles due to the two cases of generation and absorption flows. Here having higher kinetic energy with their higher velocity, temperature and nanoparticle volume friction. When they strike with the large, slow moving particles of the nanofluid they push the latter away from the boundary layer. The force that has pushed the nanoparticles away from the boundary is thermophoretic force. Due to this force enhancing the velocity and nanoparticle volume fraction whereas declines the temperature profile.

Fig. 6
figure 6

Influence of Brownian motion \((Nb)\) on velocity profile \((F^{\prime}(\varsigma ))\)

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Fig. 7
figure 7

Influence of Brownian motion \((Nb)\) on nano particle volume fraction profile \((C(\varsigma ))\)

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Fig. 8
figure 8

Influence of thermoporasis parameter \((Nt)\) on velocity profile \((F^{\prime}(\varsigma ))\)

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Fig. 9
figure 9

Influence of theroporasis parameter \((Nt)\) on temperature profile \((T(\varsigma ))\)

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Fig. 10
figure 10

Influence of thermophorasis parameter \((Nt)\) on nano particle volume fraction profile \((C(\varsigma ))\)

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Figure 11 shows the effects of the Prandtl number \(\left( {\Pr } \right)\) on the temperature profile \(T(\varsigma )\) for the two cases of heat generation and heat absorption. Also, we observed here the heat generation flow has higher thermal boundary layers thickness is compared to heat absorption flow cases. Moreover rising values of \(\left( {\Pr = { 2} . 2,\,2.4,2.6} \right)\) reduces the temperature profile. That is the small Prandtl number \((\Pr )\) shows fluids with high thermal conductivity and this creates thicker thermal boundary layer structures of large Prandtl number. Figures 12 and 13 show the consequence of Lewis number on the temperature profile, nano fluid profiles established on the heat transfer of the external flow at generation and absorption in cylindrical flow. As depicts in Fig. 12, the temperature rising very speedily and spread for small Lewis Number \(\left( {Le = 0.5,1.0,1.5} \right)\) conversely, with Lewis number boosting. The Lewis number influence on the nanoparticle volume friction profile on the convective heat transfer inside the cylinder has been shown in Fig. 13, the nanoparticle volume friction profile declines radically at the arrival region of fully established cylinder area when boosting values of Lewis number.

Fig. 11
figure 11

Influence of Prandtl number \((\Pr )\) on temperature profile \((T(\varsigma ))\)

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Fig. 12
figure 12

Influence of Lewis number \((Le)\) on temperature profile \((T(\varsigma ))\)

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Fig. 13
figure 13

Influence of Lewis number \((Le)\) on nano particle volume fraction profile \((C(\varsigma ))\)

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Tables 1, 2 and 3 displays the effects of various active physical parameters on skin fraction, Nusselt number and Sherwood numbers for with curvature and without curvature cases. From these Tables, we found that the rate of heat transfer is increased with the thermophoresis and reduced with the Brownian motion as well as heat generation parameters; these are displayed in Table 2. Here, it is interesting to mention that the friction at the wall, mass transfer rate and heat transfer rate are higher in the inclusion of curvature parameter compared to without curvature parameter. Table 1 depicts the skin friction coefficient on improving values of various physical parameters \(n,M,\lambda\). The Skin fraction coefficient has enhanced with \(M\) and decline with \(n,\lambda\). Table 3 signifies the impact of \(k_{1} ,\Pr ,Le\) on Sherwood number of curvature and without curvature in both the cases. Interestingly, we found that the Sherwood number is improved for all the parameters (Thermal Stratification, Lewis and Prandtl number).

Table 1 The modulation of skin friction for \(n,M,\lambda\) at \(Nb = 0.3,Nt, = 0.2,\)\(k_{1} = 0.2,Pr = 0.72,\)\(Le = \, 0.4.\)
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Table 2 The influence of Nusselt number for \(Nt, \, Nb\)\(, \, k_{1}\) at \(M = \, 0.6, \, n = \, 0.1,\lambda = \, 0.2, \,\) \(P \, r = \, 0.72,\)\(\, Le = \, 0.4.\)
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Table 3 The variation on Sherwood number for \(Nb, \, Nt\)\(,H^{ + }\) at \(Nt = 0.2, \, N \, b = 0.3, \,\) \(k_{1} = 0.2,\) \(M = 0.6, \, n = 0.1,\lambda = 0.2.\)
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5 Conclusion

In the present study deals with the effects of heat generation/absorption and thermal stratification on MHD Carreau nanofluid on permeable cylinder. The arising sets of nonlinear governing differential equations are converted into nonlinear ordinary differential equations with aid of appropriate similarity transformations and solved numerically. For significance of engineering interest, we also calculated the \(C_{F}\), \(Nu\) and \(Sh\). The following specified deductions are

  • The Lewis number and Thermal Stratification parameters are improves the Sherwood number for both the curvature and without curvature. It is also interesting to mention that the mass transfer rate is higher in the presence of curvature compared without curvature.

  • The Friction factor is more in the presence of curvature when compared to without curvature parameters. It clearly tells us that when we need more friction we can incorporate the curvature parameter in the flow.

  • The influence of Brownian motion is higher in nanoparticle volume fraction in the presence of magnetic field compared to without magnetic field.

  • The heat generation has higher temperature boundary compared to heat absorption case. This helps to conclude the heat absorption is useful in cooling systems.